Applications of Physics to Archery
H. O. Meyer Physics Department, Indiana University [email protected] (6 November 2015)
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Abstract: Archery lends itself to scientific analysis. In this paper we discuss physics laws that relate to the mechanics of bow and arrow, to the shooting process and to the flight of the arrow. In parallel, we describe experiments that address these laws. The detailed results of these measurements, performed with a specific bow and arrow, provide insight into many aspects of archery and illustrate the importance of quantitative information in the scientific process. Most of the proposed experiments use only modest tools and can be carried out by archers with their own equipment.
1 INTRODUCTION For more than 10,000 years, human civilizations relied on bow and arrow to provide food and to fight wars. As gunpowder gradually displaced human-powered weapons, archery declined until the 18th century when it experienced a revival as a recreational activity and as a modern sport. The behavior of bows and arrows, the shooting process, and the flight of the arrow towards the target are described and explained to a large extent by physics, mainly mechanics, elasticity and aerodynamics. Recognizing this, bowmen with scientific knowledge began to perform quantitative experiments with their bows around 1920 [1,2]. An anthology of early scientific archery papers was published as a book in 1947 [3]. Insight gained from these studies questioned the traditional longbow design and transformed bow making from a craft to a science. The continuing advancement of archery equipment, based on scientific principles, has resulted in the modern Olympic recurve bow and in the compound bow, which uses a system of cables and pulleys to modify the draw force. Crucial improvements are also due to the emergence of new plastics and compound materials, replacing traditional ingredients, such as wood, linen and animal hide. This paper, written for scientists with an interest in archery, contains a discussion of physics laws that apply to various aspects of archery and a description of experiments to test these laws. Most of these measurements require only modest tools and can be performed by readers using their own equipment. In the context of this paper, data from the proposed experiments are
1 collected for a specific bow (a compound bow) and a specific arrow, demonstrating how the understanding of many aspects of archery requires quantitative information.
2 THE ARROW
2.1 Shape, straightness and mass A modern archery arrow is shown in Fig. 1. It consists of four parts: (1) a shaft made from tubular carbon-fiber compound, aluminum, or a combination thereof, (2) the tip, or ‘pile’ made from steel or brass, sometimes screwed into an aluminum insert, (3) fletching, consisting of three or four fins of a variety of materials, shapes and sizes, and (4) a nock at the rear end of the arrow, which clips onto the bowstring and is typically made of plastic and often mounted in an aluminum insert as well. By convention, the length L of an arrow is defined as the distance from the nocking point (where the bowstring touches the arrow) to the front end of the shaft, excluding the tip. We assume that the arrow is symmetric around the z-axis of a Cartesian frame. The nocking point fixes z = 0. The y-axis shall be up and the x-axis sideways. The shaft of the arrow consists of a hollow cylinder with an outer radius R and wall thickness ΔR. The straightness of today’s carbon arrows is excellent. For instance, the axis of a moderately-priced carbon arrow is guaranteed to deviate by less than 100 μm (about the thickness of human hair) from a straight line. One can test this by rotating an arrow resting on V- notches at both ends. The wobble amplitude of the shaft in the middle, observed with a microscope, equals the deviation from straight. We have measured 10 of our sample arrows and found deviations ranging from 10 to 100 μm, with an average of 55 μm, and an accuracy of about 5 μm. This is nice to know, but it is not clear if and how such a small deviation from straight will affect the trajectory of an arrow, especially when this arrow is rotating and oscillating. A measurement of the mass M of the arrow and that of its components requires a scale with a precision of ± 0.01 g. The distribution of the mass along the sample arrow, evaluated by
weighing all parts separately, is shown in Fig. 1. From these data, the location zcm of the center of mass can be calculated. It agrees with the center of mass found by balancing the arrow on an edge. A quantity called ‘FOC’ is widely used to quantify the amount by which the center of mass is ‘Forward-Of-Center’, defined as
z FOC ≡−cm 1 . (1) L 2
It is generally accepted by the archer’s community that the FOC value should range from 0.07 to 0.17, but quantitative evidence backing up this belief seems to be lacking (see the tests with varying FOC, reported by Hickman in Ref. 3, p.77)
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Fig. 1 Linear mass distribution of the sample arrow [4]. The four parts of an arrow are indicated by numbers (see text)
Table 1. Static parameters of the sample arrow [4] L 0.696±0.001 m arrow length R 3.63±0.05 mm shaft radius ΔR 0.50±0.07 mm wall thickness M 20.62±0.01 g total arrow mass
mshaft 11.70±0.01 g (57% of total)
mtip 6.20±0.01 g (30%)
mfins 1.41±0.01 g (7%)
mnock 1.24±0.01 g (6%)
zcm 0.414±0.001 m center of mass coordinate FOC 0.093±0.002 -- ‘Forward-Of-Center’ S 5.17±0.03 Nm2 stiffness Spine 492±3 -- ATA ‘spine’
2.2 Stiffness The stiffness S is a property of the shaft material and quantifies the ability of an arrow to bend. It is of crucial importance in archery for two reasons. First, the stiffness must have a minimum value to prevent destruction of the arrow during acceleration (sects. 4.1, 4.2), and second, it has to have a specific value for the proper interaction between the bow and the arrow during the shooting process (sect. 4.3). The stiffness is easily measured as follows. In the setup shown in Fig. 2, a section of the arrow is supported by two knife edges, which allow the arrow to tilt around the x-direction. Small notches in the edges keep the arrow from rolling off sideways. The distance Δz between the two supports is arbitrary, but must be known (here, Δz = 0.550 m).
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Fig. 2 Setup to measure the stiffness of the sample arrow
A varying downward force Fb is applied halfway between the supports by hanging weights or by pulling with a luggage scale. The downward deflection ym at the mid-point is measured with a dial indicator. The measured ym as a function of Fb are shown in Fig. 3. As expected for elastic deformation, the deflection is proportional to the applied force,. The physics that governs the slope ym/Fb is understood within the Euler-Bernoulli beam theory [5] which is valid for small deflections of thin beams under lateral loads. The expected result for the present situation is
Δz3 yF≡ . (2) mb48S
This equation defines the stiffness S [Nm2]. Since all other quantities in Eq. (2) have been measured, the stiffness can be deduced (listed in table 1). The Archery Trade Association (ATA) proposes to measure stiffness by a different quantity, called Spine which is in common use. Their definition reads: “Spine is the number of thousands of an inch by which the center of an arrow shaft of 28 inch length is displaced when a sideways force of 1.94 pounds is acting at that point”. To convert stiffness S into Spine, the above definition, expressed in SI units, is inserted into Eq. 2. It follows that “Spine equals 2544 divided by the stiffness S in Nm2”. The result for our sample arrow is 492, close to the value of Spine 500, quoted by the manufacturer.
4 20
15 (mm) m
y 10
5 deflection
0 0 102030
center force Fb (N)
Fig. 3 Deflection ym versus applied force Fb, measured with the setup shown in Fig.2
It turns out that stiffness is the product of two factors, one related to the geometry and the other to the material properties of the shaft.
=⋅ SJYxxz. (3)
The geometry factor, Jxx, is the second moment of the beam cross-section relative to the x- axis. For a hollow cylinder with an outer radius of R and inner radius of R−ΔR, Jxx is given by
π J==−−Δ y244 dx dy() R() R R . (4) xx 4
─11 4 With R and ΔR from table 1, one obtains Jxx = 6.10·10 m . When we assume that ΔR is much 3 smaller than R, the moment Jxx (and thus the stiffness) is proportional to ΔR and to R . The stiffness of an arrow is therefore affected more strongly by the diameter than by the wall thickness. The other factor in Eq. (3) is the elastic modulus, which quantifies strain in response to a given stress and is a material property. The mechanical properties of carbon fiber composite materials are not well defined and even depend on direction (here we need the modulus Yz that applies in the z-direction). However, we may treat Yz in Eq.(3) as the unknown and deduce from 10 2 our data a value of Yz = 8.48 ·10 N/m , which is in line with literature values for carbon fiber materials.
5 2.3 Transverse oscillation 2.3.1 Measurement of the fundamental transverse oscillation An arrow in flight is oscillating sideways, to some degree. The dominant ‘fundamental’
mode is shown in Fig. 4. The two stationary points α and β are called nodes. Their positions (zα = 0.125 m, zβ = 0.630 m) can be calculated from a model as discussed in sect. 2.3.2. An experiment to study the fundamental mode is described in the following. As shown in Fig. 5, the arrow is supported at the nodes by mounts that constrain the arrow vertically but allow tilting around a horizontal axis (Figs. 6a and 6b). Thus, the mechanical support does not interfere with the fundamental oscillation.
Fig. 4 Fundamental mode of transverse oscillation. The coloration is related to displacement. The two nodes are labelled α and β
The oscillation is driven by a magnetic force acting on the steel tip. The driving magnet (‘2’ in Fig. 5) consists of a 200-turn coil on a 20 by 20 mm2 iron core. There is a 5 mm gap between the arrow tip and the magnet pole. The coil is energized by an alternating current of about 1 A (rms). The frequency of the current is controlled to within 0.01 Hz by a digital waveform generator. Since the magnetic force is attractive, independent of the sign of the current, the excitation frequency is twice the frequency of the current. The amplitude of the oscillating arrow is measured by slowly lowering a sensor (a 25 μm thick stainless steel reed, (‘3’), until it touches the arrow when at maximum deflection. The electrically isolated arrow is connected to a 5 V power supply (‘3’) and contact between the reed and the arrow is detected electrically. The sensor is mounted on a micrometer stage; the amplitude is deduced from the micrometer reading when contact occurs. The measurements are reproducible to within a few hundredths of a millimeter. The data acquired with the driven sample arrow cover the resonance of the fundamental mode and are shown in Fig. 7.
Fig. 5 Experiment to measure the transverse oscillation of an arrow. The numbered items are identified in the text
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Fig. 6 a), b) support at the nodes between two stretched strings: the vertical position is fixed but a tilt is allowed; c) the steel tip above the magnet. The electrical connection to the arrow is provided by the wire on the left
From the stiffness data in Fig. 3, we know that the restoring force is proportional to the displacement. This is the signature of a harmonic oscillator. The theoretical treatment of a driven
harmonic oscillator yields the following equation for the mid-point amplitude ym as a function of the driving frequency f ,
y 0 = m yfm () , (5) +−()()2 121Qff0
0 where f0 is the resonance frequency and ym is the amplitude on resonance. The ‘Q factor’ is proportional to the rate of energy loss due to friction. The curve in Fig. 7 has been calculated
with Eq. (5), adjusting f0 and Q for best fit. The resulting resonance frequency is
=± . (6) f 0,exp (84.8 0.1)Hz
For the Q-factor we find Q = 400 ± 30. If this parameter is large, the frictional energy loss in the system is low and the resonance is narrow. The damping time, τ1/e , during which a free oscillation decreases to 1/e = 0.37 of its initial amplitude, is given by
Q τ = . (7) 1/e π f 0
With our values for Q and f0, the damping time amounts to τ1/e ~ 1.5 s. This is longer than the flight time of an arrow in most target archery situations. Thus, when the arrow hits the target, most of the initial transverse oscillation will still be present, if aerodynamic damping can be neglected.
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Fig. 7 Transverse oscillation amplitude of the sample arrow versus excitation frequency (solid dots). The curve represents
Eq. (5) with the parameters f0 = 84.4 Hz and Q = 400
2.3.2 Finite-element modelling Advances in computing power and software have made it possible to simulate the dynamics of physical objects numerically. Of extensive use in this context is the finite-element (FE) method. This is a technique for finding approximate solutions to boundary-value problems represented by partial differential equations. To this aim, the structure under study is sub-divided into small volume elements. In each element the physics problem is solved exactly and then the elements are matched. The FE model used here is the 3D Structural Mechanics Module of the commercial software COMSOL [6] Similar computer modelling studies of arrows have been reported in the literature [7]. The arrow and its parts are defined in the program with some simplifications in shape but with masses that match those of their real counterparts exactly. Young’s modulus Yz of the shaft material is chosen such that for the given moment of inertia Jxx, the measured stiffness is reproduced. Thus, all known static properties are represented in the FE model. Constraints and external forces are turned off, so the calculation models free motion. Among the quantities that COMSOL calculates are the coordinates of the two nodes, = 0.125 m and = 0.630 m and the frequency of the fundamental mode, f0,C = 85.4 Hz. The excellent agreement between the calculated and the measured frequency, Eq.(6), suggests that one may use the model with confidence to calculate other parameters and to make predictions. For example, the model knows the kinetic energy for each element and thus the time-averaged kinetic oscillation energy of the entire object. The total energy is the sum of potential and kinetic energy and, averaged over time, the two are equal. Using the fact that is proportional to the square of the amplitude of the central displacement, ym, we obtain = 2 1080 J/m · for our arrow. For instance, with a relatively large amplitude of ym = 2 cm, the
8 oscillation energy is ~ 0.43 J. We note that the oscillation energy is small compared to the kinetic energy of the moving arrow. This will be discussed further in sect. 3.3.2. The FE model is also useful in predicting the effect of changes when considering modifications of a given arrow. For instance, when the length L of the sample arrow is decreased
by 5 cm (by 7%), the frequency f0 increases to 96.0 Hz (by 13%). When the stiffness S is increased by 20% (by lowering the ‘spine’ from 500 to 400), the frequency increases to 95.0 Hz (by 12%). The FE code also calculates the higher resonant modes, which are probably unimportant in archery. For example, the second mode occurs at 272 Hz and has three nodes at z = 0.070 m, 0.365 m, and 0.670 m.
2.3.3 Theory of vibrating beams The Euler-Bernoulli theory [5], mentioned earlier, can be formulated to describe the time- dependence of the motion of beams. This leads to closed-form expressions for performance parameters, such as the resonance frequency, in terms of the properties of the system. Simplifying our arrow as a thin, uniform, free, vibrating beam of stiffness S, length L and linear mass density μ' , we arrive at the following equation of motion
∂∂42y(,)zt μ′ y (,)zt =− . (8) ∂∂zSt42
This equation relates the changes of the displacement from the axis, y, with position z along the arrow and with time t. The length L of the beam enters through boundary conditions. The solutions of Eq. (8) are discrete, associated with the modes of oscillation. They describe the shape of the bent beam (in particular the location of the nodes) and the frequency of a given mode. The result for the frequency of the fundamental mode is
1 S c 2 = 0 . (9) f0,th 2πμ′ L
The constant c0 = 4.730 is the smallest (discrete) solution of the boundary equation cos( ) ∙ cosh( ) =1. For the linear mass density μ' of the idealized beam, we average the arrow mass over its entire length, μ ′ = M / L ; values for M, L, and S are from table 1. The result, f0,th = 97.2
Hz, differs from the measured value, f0,exp = (84.8 ± 0.1) Hz (Eq. (6)), by quite a bit. Nevertheless, Eq. (9) is still very informative because it tells us how the free oscillation frequency depends on length, stiffness and density. For example, if we wanted to increase the frequency, say, by 10%, we would have to decrease the arrow length by only 5%, but we would have to increase the stiffness by 20%. Moreover, we see that if a stiffness increase were
9 accompanied by an increase of the linear density (which is likely to be the case), the two changes would tend to compensate each other.
3 THE BOW The task of the bow is to store the mechanical energy that is produced by the archer. A good bow should, when triggered, transfer as much of the potential bow energy as possible into kinetic energy of the arrow. Some components of the bow [8] are identified in Fig. 8. The rigid center part, made from cast aluminum, is the riser. Attached to it are the elastic limbs, which bend when the bow is drawn and store energy. The grip is where the hand of the archer pushes while shooting. The point on the string where the arrow nock is placed in preparation for the shot is the nocking point. Typical for a compound bow are the extra cables and the cams (off-center pulleys).
3.1 Measuring the stored energy The setup to measure the draw force is shown in Fig. 8. The bow is tied down at the grip and
pulled up by the force Fd , applied at the nocking point in the direction indicated at the top of the figure. This ‘draw force’ is a function of the distance s between the grip and the nocking point.
For a relaxed bow this distance equals s0, called brace height. While the bow is drawn, the nocking point moves by a distance sD, and at full draw s = s0 + sD, the true draw length. For our sample bow [8] s0 = 0.185 m and sD = 0.481 m. During the measurement, the bow is supported just at the grip and at the nocking point and can rotate around a straight line through these two points. The wooden bar on the left is there to prevent this. The force is applied by a rope attached to the nocking point. A block and tackle (not shown) is used to help pulling the rope. The force is measured by a piezo-electric load cell and is displayed on a read-out at the bottom of the picture.
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Fig. 8 Definition of some bow terms and setup to measure the draw force as a function of the distance s. In this picture, s = 0.432 m
To determine the amount of stored energy, we need to know the draw force Fd as a function of the drawn distance s. To measure this we draw the bow in small steps, taking a picture after every step. To reduce parallax, a 200 mm lens at 10 m is used. After some 15 exposures, having reached full draw, this process is reversed, until back to the rest position. Consequently, data are obtained when drawing the bow ( ( )) and when releasing it ( ( )). The draw length s is extracted by digitizing [9] the coordinates of the points of interest in each picture. A bow-fixed coordinate frame with an absolute length scale is defined, based on the two reference points on the riser, marked by red circles. The data are shown in Fig. 9. It is typical for a compound bow that the draw force first rises steeply, then is roughly constant over much of the draw and then decreases sharply. In the present case this ‘let-off’ at full draw is about 65 %. The sharp increase beyond full draw occurs because the cables run out of length. The same graph would look much different for a conventional bow, where the draw force increases monotonically over the entire draw.
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